CSCI 6212 Design and Analysis of Algorithms
Which algorithm is better ?
Dr. Juman ByunThe George Washington University
Please drop this course if you have not taken the following prerequisite.
Sometimes enthusiasm alone is not enough.
• CSci 1311: Discrete Structures I (3)• CSci 1112: Algorithms and Data Structures
(3)
Running Time Calculation
Example: Running Time Analysis
Insertion Sort (A)1 for j = 2 to A.length2 key = A[j]3 // Insert A[j] into the sorted sequence A[1..j-1]4 i = j - 15 while i > 0 and A[i] > key6 A[i +1] = A[i]7 i = i - 18 A[i + 1] = key
LineLine CostCost TimeTimess
11 c1 n
22 c2 n-1
33 c3 n-1
44 c4 n-1
55 c5
66 c6
77 c7
88 c8 n-1
LineLine CostCost TimeTimess
11 c1 n
22 c2 n-1
33 c3 n-1
44 c4 n-1
55 c5
66 c6
77 c7
88 c8 n-1
Best Case: already sorted
Insertion Sort (A)1 for j = 2 to A.length2 key = A[j]3 // Insert A[j] into the sorted sequence A[1..j-1]4 i = j - 15 while i > 0 and A[i] > key6 A[i +1] = A[i]7 i = i - 18 A[i + 1] = key
Best Case: already sorted
Insertion Sort (A)1 for j = 2 to A.length2 key = A[j]3 // Insert A[j] into the sorted sequence A[1..j-1]4 i = j - 15 while i > 0 and A[i] > key6 A[i +1] = A[i]7 i = i - 18 A[i + 1] = key
Best Case: already sorted
Best Case: already sorted
• simply express it
Worst Case: reverse sorted
Insertion Sort (A)1 for j = 2 to A.length2 key = A[j]3 // Insert A[j] into the sorted sequence A[1..j-1]4 i = j - 15 while i > 0 and A[i] > key6 A[i +1] = A[i]7 i = i - 18 A[i + 1] = key
Worst Case: reverse sorted
Insertion Sort (A)1 for j = 2 to A.length2 key = A[j]3 // Insert A[j] into the sorted sequence A[1..j-1]4 i = j - 15 while i > 0 and A[i] > key // for entire A[1..j-1]6 A[i +1] = A[i] //∴tj = (j - 1) + 17 i = i - 1 // = j8 A[i + 1] = key
Worst Case: reverse sorted
Worst Case: reverse sorted
Worst Case: reverse sorted
Worst Case: reverse sorted
Worst Case: reverse sorted
Worst Case: reverse sorted
Worst Case: reverse sorted
Worst Case: reverse sorted
Worst Case: reverse sorted
• To abstract running time T(n) using constants
• Notation of "Worst-Case Running Time of Insertion Sort"
Worst Case: reverse sorted
• Simply abstract it using constants
To Denote Relative Algorithm Performance
• Algorithm 1 input size n with running time f(n)
• Asymptotic Notation
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